Phase problem molecular replacement

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

One major problem crystallographers have to deal with is the so-called phase problem, which states that of the two components of an irrational Figure (magnitude and phase) only the magnitude can be measured. A technique called molecular replacement is an approach to deal with this problem [131]. 

The possibility and feasibility of molecular replacement was demonstrated by Rossmann and colleagues in the 1960s, as part of an effort to use non-crystallographic synnmetry to solve the phase problem for macromolecules (Rossmann, 1990). 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

An alternative procedure, called molecular replacement, uses information about known structures that are believed to be similar to that of the species being investigated. The known structure is used to estimate the electron density of the unknown structure, which is then refined and improved. Another method of dealing with the phase problem is to introduce atoms which absorb radiation in the region of the incident X-rays, leading to a process called anomalous scattering . For proteins, a popular method is to replace S by Se by using selenomethionine in place of methionine. For nucleic acids, iodouracil or iodocytosine can be used in place of thymine and cytosine respectively. 

Simulated annealing has also proven useful with the reciprocal space equivalent of the real space search problem. More conventionally known as the molecular replacement problem, this reciprocal space search problem occurs when at the outset of the crystallographic experiment a reasonably detailed approximate model of the macromolecule is already available and the intention is to altogether avoid the painful acquisition of heavy-atom phases (Rossman, 1972). The absence of heavy-atom derived phases differentiates this reciprocal space version of the search problem from its real space analog previously discussed. 

In these instances, the need for heavy atoms can be frequently side-stepped by performing a reciprocal space search in 6 dimensions - 3 rotational and 3 translational. For each point in this vast 6-dimensional space, the calculated Fourier amplitudes from the suitably rotated and translated model can be compared with the experimental Fourier amplitudes. Such an exhaustive search can in principle give the correct orientation and location of the available approximate model in the new crystal. This allows the calculation of approximate phases for the crystal structure and ultimately leads to an accurate atomic structure. However, such a molecular replacement solution does not always work. This is because in practice, a truly exhaustive 6-dimensional search is not possible given present day computing resource. So this 6-dimensional problem is routinely split into two far smaller and consecutive 3-dimensional problems - 3-dimensional... 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

FIGURE 30.11 The phase problem. The experimental data obtained in an X-ray experiment are the intensities of the reflections. By using an inverse Fourier transform, it is possible to calculate electron-density maps from these intensities. However, it is essential for this calculation to know the phase associated with each reflection. Approximate initial phases can be obtained from heavy-atom derivatives, anomalous dispersion or molecular replacement (see text). More accurate phases can be derived from the refined model, once it has been obtained. 

Molecular replacement. When a suitable model of the unknown crystal structure is available, it can be used to solve the phase problem. Examples are the use of the structure of human thrombin to solve the structure of bovine thrombin, the use of a known antibody fragment to solve the sttucture of an unknown antibody, or the use of the stmcture of an enzyme to solve the structure of an inhibitor complex in a different crystal form. The model is oriented and positioned in the unit cell of the unknown crystal with the use of rotation and translation functions, and the oriented model is subsequently used to calculate phases and an electron density map. 

The application of molecular replacement relies on similar model proteins. If there are no known proteins that are similar to the proteins we want to study, then in such cases, what we face are the de novo structures, and molecular replacement is not valid any more. Isomorphous replacement or anomalous scattering methods have to apply in order to solve the phase problems. 

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

The use of TFA as a mobile-phase additive in LC-MS can be problematical when using electrospray ionization. In negative ion detection, the high concentration of TFA anion can suppress analyte ionization. In positive ion detection, TFA forms such strong ion pairs with peptides that ejection of peptide pseudo-molecular ions into the gas phase is suppressed. This problem can be alleviated by postcolumn addition of a weaker, less volatile acid such as propionic acid.14 This TFA fix allows TFA to be used with electrospray sources interfaced with quadrupole MS systems. A more convenient solution to the TFA problem in LC-MS is to simply replace TFA with acetic or formic acid. Several reversed-phase columns are commercially available that have sufficient phase coverage and reduced levels of active silanols such that they provide satisfactory peptide peak shapes using the weaker organic acid additives.15... 

Continuum models are the most efficient way to include condensed-phase effects into quantum-mechanical calculations, and this is typically accomplished by using the self-consistent reaction field (SCRF) approach for the electrostatic component. Therefore it is very common to replace the quantal problem by a classical one in which the electronic energy plus the coulombic interactions of the nuclei, taken together, are modeled by a classical force field—this approach usually called molecular mechanics (MM) (Cramer and Truhlar, 1996). 

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